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Theorem signslema 28159
Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
Hypotheses
Ref Expression
signslema.1  |-  ( ph  ->  E  e.  NN0 )
signslema.2  |-  ( ph  ->  F  e.  NN0 )
signslema.3  |-  ( ph  ->  G  e.  NN0 )
signslema.4  |-  ( ph  ->  H  e.  NN0 )
signslema.5  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
signslema.6  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
Assertion
Ref Expression
signslema  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )

Proof of Theorem signslema
StepHypRef Expression
1 signslema.5 . . . . . 6  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
21simpld 459 . . . . 5  |-  ( ph  ->  E  <  G )
32adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  E  <  G )
4 signslema.4 . . . . . . . . . 10  |-  ( ph  ->  H  e.  NN0 )
54nn0cnd 10850 . . . . . . . . 9  |-  ( ph  ->  H  e.  CC )
6 signslema.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  NN0 )
76nn0cnd 10850 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
85, 7subcld 9926 . . . . . . . 8  |-  ( ph  ->  ( H  -  F
)  e.  CC )
9 signslema.3 . . . . . . . . . 10  |-  ( ph  ->  G  e.  NN0 )
109nn0cnd 10850 . . . . . . . . 9  |-  ( ph  ->  G  e.  CC )
11 signslema.1 . . . . . . . . . 10  |-  ( ph  ->  E  e.  NN0 )
1211nn0cnd 10850 . . . . . . . . 9  |-  ( ph  ->  E  e.  CC )
1310, 12subcld 9926 . . . . . . . 8  |-  ( ph  ->  ( G  -  E
)  e.  CC )
148, 13subeq0ad 9936 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  <-> 
( H  -  F
)  =  ( G  -  E ) ) )
1514biimpa 484 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( H  -  F )  =  ( G  -  E ) )
1615breq2d 4459 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
0  <  ( H  -  F )  <->  0  <  ( G  -  E ) ) )
176nn0red 10849 . . . . . . 7  |-  ( ph  ->  F  e.  RR )
184nn0red 10849 . . . . . . 7  |-  ( ph  ->  H  e.  RR )
1917, 18posdifd 10135 . . . . . 6  |-  ( ph  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2111nn0red 10849 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
229nn0red 10849 . . . . . . 7  |-  ( ph  ->  G  e.  RR )
2321, 22posdifd 10135 . . . . . 6  |-  ( ph  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2516, 20, 243bitr4rd 286 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  F  <  H ) )
263, 25mpbid 210 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  F  <  H )
27 0re 9592 . . . . . 6  |-  0  e.  RR
2827a1i 11 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  e.  RR )
2922, 21resubcld 9983 . . . . . 6  |-  ( ph  ->  ( G  -  E
)  e.  RR )
3029adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  e.  RR )
3118, 17resubcld 9983 . . . . . 6  |-  ( ph  ->  ( H  -  F
)  e.  RR )
3231adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( H  -  F )  e.  RR )
332adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  E  <  G )
3423adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
3533, 34mpbid 210 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( G  -  E
) )
36 2pos 10623 . . . . . . . 8  |-  0  <  2
37 breq2 4451 . . . . . . . 8  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  (
0  <  ( ( H  -  F )  -  ( G  -  E ) )  <->  0  <  2 ) )
3836, 37mpbiri 233 . . . . . . 7  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
3938adantl 466 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
4029, 31posdifd 10135 . . . . . . 7  |-  ( ph  ->  ( ( G  -  E )  <  ( H  -  F )  <->  0  <  ( ( H  -  F )  -  ( G  -  E
) ) ) )
4140biimpar 485 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( H  -  F
)  -  ( G  -  E ) ) )  ->  ( G  -  E )  <  ( H  -  F )
)
4239, 41syldan 470 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  <  ( H  -  F
) )
4328, 30, 32, 35, 42lttrd 9738 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( H  -  F
) )
4419adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
4543, 44mpbird 232 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  F  <  H )
465, 10, 7, 12sub4d 9975 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  =  ( ( H  -  F )  -  ( G  -  E ) ) )
47 signslema.6 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
4846, 47eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( ( H  -  F )  -  ( G  -  E )
)  e.  { 0 ,  2 } )
49 ovex 6307 . . . . 5  |-  ( ( H  -  F )  -  ( G  -  E ) )  e. 
_V
5049elpr 4045 . . . 4  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  e.  { 0 ,  2 }  <->  ( (
( H  -  F
)  -  ( G  -  E ) )  =  0  \/  (
( H  -  F
)  -  ( G  -  E ) )  =  2 ) )
5148, 50sylib 196 . . 3  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  \/  ( ( H  -  F )  -  ( G  -  E
) )  =  2 ) )
5226, 45, 51mpjaodan 784 . 2  |-  ( ph  ->  F  <  H )
531simprd 463 . . . . 5  |-  ( ph  ->  -.  2  ||  ( G  -  E )
)
5453adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( G  -  E ) )
5515breq2d 4459 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
2  ||  ( H  -  F )  <->  2  ||  ( G  -  E
) ) )
5654, 55mtbird 301 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( H  -  F ) )
57 2z 10892 . . . . . . 7  |-  2  e.  ZZ
589nn0zd 10960 . . . . . . . 8  |-  ( ph  ->  G  e.  ZZ )
5911nn0zd 10960 . . . . . . . 8  |-  ( ph  ->  E  e.  ZZ )
6058, 59zsubcld 10967 . . . . . . 7  |-  ( ph  ->  ( G  -  E
)  e.  ZZ )
61 dvdsaddr 13880 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  ( G  -  E
)  e.  ZZ )  ->  ( 2  ||  ( G  -  E
)  <->  2  ||  (
( G  -  E
)  +  2 ) ) )
6257, 60, 61sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  ||  ( G  -  E )  <->  2 
||  ( ( G  -  E )  +  2 ) ) )
6353, 62mtbid 300 . . . . 5  |-  ( ph  ->  -.  2  ||  (
( G  -  E
)  +  2 ) )
6463adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( ( G  -  E )  +  2 ) )
65 2cnd 10604 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
668, 13, 65subaddd 9944 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  2  <-> 
( ( G  -  E )  +  2 )  =  ( H  -  F ) ) )
6766biimpa 484 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
( G  -  E
)  +  2 )  =  ( H  -  F ) )
6867breq2d 4459 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
2  ||  ( ( G  -  E )  +  2 )  <->  2  ||  ( H  -  F
) ) )
6968notbid 294 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( -.  2  ||  ( ( G  -  E )  +  2 )  <->  -.  2  ||  ( H  -  F
) ) )
7064, 69mpbid 210 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( H  -  F ) )
7156, 70, 51mpjaodan 784 . 2  |-  ( ph  ->  -.  2  ||  ( H  -  F )
)
7252, 71jca 532 1  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {cpr 4029   class class class wbr 4447  (class class class)co 6282   RRcr 9487   0cc0 9488    + caddc 9491    < clt 9624    - cmin 9801   2c2 10581   NN0cn0 10791   ZZcz 10860    || cdivides 13843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-dvds 13844
This theorem is referenced by: (None)
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